Abstract:
The paper investigates an inverse problem for a second-order parabolic equation in a two-dimensional domain with variable time direction. The objective is to determine an unknown source function together with the solution, subject to boundary, initial, final, and gluing conditions. The separation of variables method reduces the problem to a spectral formulation involving eigenvalues and eigenfunctions. Using orthogonality, explicit series expansions for the solution and source are derived. Convergence of the series is shown, while existence and uniqueness of a classical solution are established via functional analysis and the Hilbert–Schmidt theorem.
Keywords:inverse problem, mixed-type differential equations, separation of variables, spectral problem, eigenvalues, eigenfunctions, orthogonality, existence and uniqueness
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